We study the evolution of galactic bars and the link with disk and spheroid formation in a sample of zoom-in cosmological simulations. Our simulation sample focuses on galaxies with present-day stellar masses in the 1010-1011 M ☉ range, in field and loose group environments, with a broad variety of mass growth histories. In our models, bars are almost absent from the progenitors of present-day spirals at z > 1.5, and they remain rare and generally too weak to be observable down to z 1. After this characteristic epoch, the fractions of observable and strong bars rise rapidly, bars being present in 80% of spiral galaxies and easily observable in two thirds of these at z ≤ 0.5. This is quantitatively consistent with the redshift evolution of the observed bar fraction, although the latter is presently known up to z 0.8 because of band-shifting and resolution effects. Our models hence predict that the decrease in the bar fraction with increasing redshift should continue with a fraction of observable bars not larger than 10%-15% in disk galaxies at z > 1. Our models also predict later bar formation in lower-mass galaxies, in agreement with existing data. We find that the characteristic epoch of bar formation, namely redshift z 0.8-1 in the studied mass range, corresponds to the epoch at which today's spirals acquire their disk-dominated morphology. At higher redshift, disks tend to be rapidly destroyed by mergers and gravitational instabilities and rarely develop significant bars. We hence suggest that the bar formation epoch corresponds to the transition between an early "violent" phase of spiral galaxy formation at z ≥ 1 and a late 'secular' phase at z ≤ 0.8. In the secular phase, the presence of bars substantially contributes to the growth of the (pseudo-)bulge, but the bulge mass budget remains statistically dominated by the contribution of mergers, interactions, and disk instabilities at high redshift. Early bars at z > 1 are often short-lived, while most of the bars formed at z ≤ 1 persist down to z = 0, late cosmological gas infall being necessary to maintain some of them.